Multiplying and Dividing Integers

Once you are comfortable adding and subtracting integers, multiplication and division are surprisingly straightforward. The arithmetic itself is the same as with whole numbers, and there is just one new piece: the sign rules. These rules tell you whether the answer is positive or negative based on the signs of the numbers you are working with.

The Sign Rules

The sign rules for multiplication and division are identical:

The pattern is simple:

$\text{Same signs} \rightarrow \text{positive result}$

$\text{Different signs} \rightarrow \text{negative result}$

These same rules apply to division:

$\text{Same signs} \rightarrow \text{positive quotient}$

$\text{Different signs} \rightarrow \text{negative quotient}$

Why Does Negative Times Negative Equal Positive?

This is one of the most common questions in arithmetic, and there are several ways to understand it.

Pattern approach: Look at the pattern as you multiply $-3$ by decreasing numbers:

$-3 \times 3 = -9$ $-3 \times 2 = -6$ $-3 \times 1 = -3$ $-3 \times 0 = 0$ $-3 \times (-1) = \;?$

Each time the second number decreases by 1, the product increases by 3. Following the pattern:

$-3 \times (-1) = 3$ $-3 \times (-2) = 6$ $-3 \times (-3) = 9$

The pattern demands that a negative times a negative is positive.

Opposite approach: Multiplying by $-1$ gives you the opposite of a number. The opposite of $-3$ is $3$:

$(-1) \times (-3) = 3$

Worked Examples: Multiplication

Example 1: Positive Times Positive

$7 \times 4 = 28$

Same signs (both positive), so the result is positive. No surprises here.

Example 2: Negative Times Positive

$(-5) \times 3 = -15$

Different signs, so the result is negative. Multiply the absolute values ($5 \times 3 = 15$) and attach a negative sign.

Example 3: Positive Times Negative

$6 \times (-8) = -48$

Different signs, so the result is negative. Multiply: $6 \times 8 = 48$. Result: $-48$.

Example 4: Negative Times Negative

$(-9) \times (-4) = 36$

Same signs (both negative), so the result is positive. Multiply: $9 \times 4 = 36$.

Example 5: Three or More Factors

When multiplying more than two integers, count the negative signs:

• Even number of negatives: positive result.
• Odd number of negatives: negative result.

$(-2) \times (-3) \times (-5) = ?$

Three negative factors (odd count), so the result is negative. Multiply the absolute values: $2 \times 3 \times 5 = 30$. Result: $-30$.

$(-2) \times (-3) \times (-5) = -30$

Worked Examples: Division

Division follows the same sign rules as multiplication.

Example 6: Positive Divided by Positive

$\frac{24}{6} = 4$

Same signs, positive result.

Example 7: Negative Divided by Positive

$\frac{-36}{4} = -9$

Different signs, negative result. Divide the absolute values: $36 \div 4 = 9$. Attach the negative sign.

Example 8: Positive Divided by Negative

$\frac{45}{-9} = -5$

Different signs, negative result.

Example 9: Negative Divided by Negative

$\frac{-56}{-8} = 7$

Same signs, positive result. Divide: $56 \div 8 = 7$.

Multiplication by Zero and One

Two special cases worth noting:

• Any integer times zero is zero: $(-7) \times 0 = 0$
• Any integer times one is itself: $(-7) \times 1 = -7$
• Any integer times negative one is its opposite: $(-7) \times (-1) = 7$

Also, division by zero is undefined. You cannot divide any integer by zero.

Practice Problems

Problem 1: $(-8) \times 7$

Problem 2: $(-12) \times (-5)$

Problem 3: $\frac{-72}{-9}$

Problem 4: $(-3) \times (-2) \times 4 \times (-1)$

Problem 5: $\frac{100}{-25}$

Key Takeaways

• Same signs (both positive or both negative) produce a positive result for both multiplication and division.
• Different signs (one positive, one negative) produce a negative result.
• For products with three or more factors, count the negative signs: an even count gives a positive result; an odd count gives a negative result.
• Negative times negative equals positive, which you can verify through patterns or the concept of opposites.
• Any number times zero is zero. Division by zero is undefined.
• The sign rules for division are exactly the same as for multiplication.

Return to Arithmetic for more foundational math topics.